For integers $n\geq 1$, let $a:=\left\{ a(n) \right\}_{n=1}^\infty $ and $b:= \left\{ b(n) \right\}_{n=1}^\infty $ fixed sequences of integers with general terms $a(n)\geq 2$ and $b(n)\geq 2$. And additionally we consider that both sequences are strictly increasing.
Then we consider the sequence $$P_a^b(n)=P(n):=\frac{(a(n))^{b(n)}-1}{a(n)-1}=1+a(n)+\ldots+(a(n))^{b(n)-1}.\tag{1}$$
Example. An example of such sequence $P_a^b(n)$ is take $a(n)=p_n$, where $p_n$ is the $nth$ prime number and $b(n)=n+2$, for $n\geq 2$.$\square$
We denote the greatest prime factor of an integer $n\geq 2$ with $\operatorname{gpf}(n)$.
Since we can divide $(1)$ by $a(n)$, one can consider in some way that $P(n)$ is a generator of primes.
Question 1. How to specify that our sequence $P(n)=P_a^b(n)$ is rich in prime numbers? Many thanks.
That is, I don't know if this question was in the literature, if there are special sequences $a(n)$ and $b(n)$ satisfying our requirements and such that $P(n)$ is rich in primes in some way: well because $P(n)$ is itself a prime number for many $n's$ or well because there is abundance of distinct primes in the sequence $$\operatorname{gpf}(P_a^b(n)),$$ for $n\geq 1$.
Notice that the sequence in Euclid's proof for the infinitude of primes has the form $$b(n)= 2$$ for all $n\geq 1$, and $a(n)=N_n$ the primorial of order $n$.
Question 2. Then, after you've answered previous question, answer this new: Were in the literature special choices of sequences $a$ and $b$ such that the sequence $P_a^b(n)$ is rich in primes? Then refers it, and I try find such literature. In other case if you know how create an example of sequences $a$ and $b$ satisfying the requirements of our first paragraph, and such that the sequence $P_a^b(n)$ is extraordinarily rich in primes show us your example. Many thanks.